An exact solution for the motion of a slender body sweeping a cone in a viscous fluid (Terry Jo Leiterman, UNC-Chapel Hill)

Abstract: The uniform motion of a thin body in a viscous (Stokes) fluid is a classical moving boundary problem which has been well studied using slender body asymptotic methods. The drag law and associated fluid flow induced by this motion has been deduced using an array of fundamental singularities of the Stokes equations known as Stokeslets. However, for non-uniform motions, much less is known. In 1970, Batchelor extended the slender body theory to allow for linear, non-uniform far field flow boundary conditions using only Stokeslets singularities. It is well known that slender body theory suffers near the ends of the body, but for uniform far field flow conditions, the method is asymptotic. By extending the family of singularities, we show how to correct for this, and, present an asymptotic correction for slender body theory under linear, far field flow conditions, which is, relevant for any sweeping motion of a rigid body. In addition, we present a new exact solution of the Stokes equations for a spinning rod sweeping a cone. These solutions are being compared to micro-fluidic mixing experiments performed using a 3D magnetic force microscope developed by Rich Superfine and collaborators in physics at UNC. We anticipate these new solutions and asymptotic methods to play a fundamental role in the study of transport in ciliated tissues. This is joint work with R. Camassa and R. McLaughlin.

---

This is an abstract of a talk to be presented at the 2004 SEAMS Workshop in Charleston, SC. For more information, visit the workshop's homepage at math.cofc.edu/SEAMS.